**A Thesis Submitted for the Degree of PhD at the University of Warwick**

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### Resolution improvement methods applied to

### digital holography

### Daniel Claus

### Dipl.-Ing.

This report is submitted as partial fulfilment of the requirements for the PhD Programme of the

School of Engineering University of Warwick

www.warwick.ac.uk

TITLE: Resolution improvement methods applied to digital holography

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### Abstract

This thesis discusses the creation, acquisition and processing of digital holograms.

Several techniques to improve the optical resolution have been investigated and

developed. The optical resolution of numerically reconstructed digital holograms

is restricted by both the sampling frequency and the overall sensor-size of the

digital camera chip used. This thesis explores the limitations on the optical

res-olution of the holograms obtained. A typical sensor-size and sampling frequency

for digital holograms is 10 mm and 100 lp/mm, respectively, whereas holographic

plates used for optical holography can be more than a meter in size and have a

sampling frequency of 3000 lp/mm. In order to take full advantage of the benefits

digital holography offers, such as fast image acquisition and direct phase

accessi-bility, the problem of reduced resolution needs to be overcome. Three resolution

improvement methods have been developed in the scope of this PhD thesis. Prior

to implementing the resolution improvement methods, different holographic

se-tups have been analyzed, using the Space-bandwidth product (SBP) to calculate

the information distribution both in the recording and reconstruction process.

The first resolution improvement method is based on the synthetic aperture

method. In this manner an increased sensor area can be obtained resulting in

a larger numerical aperture (NA). A larger NA enables a more detailed

recon-struction. The problem encountered in doing this is that an increased optical

sis by applying the extended depth of field method. As a result a high resolution

in focus reconstruction of all longitudinal object regions is obtained. Moreover,

the extended depth of field method allows a topological mapping of the object.

The second resolution improvement method is based on sampling the

inter-ference pattern with sub-pixel accuracy. This was carried out on a CMOS-sensor

and implemented by moving the light sensitive pixel-area into the dead zone in a

4x4 grid to cover whole the pixel-area. As a result the sensor’s sampling frequency

is doubled. The increased sampling frequency permits a reduction of the

record-ing distance which results in an increased optical resolution of the reconstructed

hologram.

The third and novel approach described in this thesis has been to increase

the optical resolution stored in a digital hologram by the combination of the

syn-thetic aperture and the sub-pixel sampling methodBy analogy with the Fresnel-.

The resolution improvement methods have been demonstrated both for lens-less

digital holography and digital holographic microscopy.

Keywords: digital holography, space-bandwidth product, high-resolution, syn-thetic aperture, sub-pixel, Fourier-hologram, lens-less holography, digital

holo-graphic microscope

### List of Publications

### Posters

◦ Daniel Claus, Dr. Brenda Timmerman, Prof. Peter Bryanston-Cross (2008).

Digital Holographic Interferometry. Poster Competition. University of Warwick: May 2008.

### Presentations

◦ D. Claus, P. Bryanston-Cross, B. Timmerman (2007). Digital Double

Ex-posure Holography by means of Polarization Optics. Digital Holography. Vancouver, OSA: June 2007.

◦ D. Claus, P. Bryanston-Cross, B. Timmerman (2007). Digital holography.

Colloquium School of Engineering, University of Warwick: December 2007.

◦ D. Claus, P. Bryanston-Cross, B. Timmerman (2008) Digitale Holografie.

Research Visit, Technische Universit¨at Ilmenau: June 2008.

◦ D. Claus, P. Bryanston-Cross, B. Timmerman (2008) Digital Holography.

Research Visit, Universidad de Vigo: October 2008.

lution for digital holograms. Research Visit, Institute f¨ur technische Optik

Universit¨at Stuttgart: December 2008.

◦ D. Claus, B. Timmerman, and P. Bryanston-Cross (2009). Resolution

Im-provement in Digital Holography. Colloquium Warwick Innovative Manu-facturing Research Centre, University of Warwick: April 2009.

◦ D. Claus, Marco Fritzsche, B. Timmerman, and P. Bryanston-Cross (2009).

Res-olution improvement in lensless digital holographic interferometry. Stuttgart, Fringe 2009: September 2009.

### Scientific publications

◦ D. Claus, P. Bryanston-Cross,B. Timmerman (2007). Digital Double

Ex-posure Holography by means of Polarization Optics. Digital Holography. Vancouver, OSA: June 2007.

◦ D. Claus, Marco Fritzsche, B. Timmerman, and P. Bryanston-Cross (2009).

Resolution improvement in lensless digital holographic interferometry. Stuttgart, Fringe 2009: September 2009.

◦ D. Claus (2010). High resolution digital holographic synthetic aperture

ap-plied to deformation measurement and extended depth of field method. Appl. Opt., 49(16):3187-3198, 2010

### Declaration of Originality

The author wishes to declare that apart from commonly understood and accepted

ideas, or where reference is made to the work of others, the work in this thesis is

his own. It has not been submitted in part, or in whole, to any other university

for a degree, diploma or other qualification.

Daniel Claus

Coventry, 6th _{of October 2010}

I would like to offer my sincere thanks to those people whose assistance and

support was vital to the development and success of this project.

I would like firstly to give my special thanks to my family, especially my wife

and my daughter, who gave me the necessary distance to my research project in

order to remain focused and open minded to new ideas.

Furthermore, I would like to thank my supervisors Prof. Peter

Bryanston-Cross and Dr. Brenda Timmerman, for making this project possible in

conjunc-tion with the Warwick Innovative Manufacturing Research Centre (WIMRC), for

their trust, professional advice and assistance during this research.

I would also like to thank Dr. Daciana Udrea from the University of Warwick,

Mr. Thomas Meinecke from the Technische Universit´at Ilmenau, Dr. ´Angel F.

Doval from the Universidad de Vigo, Dr. Giancarlo Pedrini from the Universit´at

Stuttgart, Prof. Ferenc Gy´ımesi from the Budapest University of Technology

and Economics, Dr. H´ector H. Cerecedo N´u˜nez and Patrica Padilla Sosa from

the Universidad de Veracruzana and Dr. Ulf Schnars from Airbus Deutschland

GmbH for their valuable discussions.

Moreover, I would like to express my gratitude to the School of Engineering

technicians. At this point I would especially like to thank Mr. Charles Joyce,

whose help and flexibility was essential when building the digital holographic

microscope. In this context I would also like to thank Dr. Robert W. Old

from the biological science department, who provided a prepared cheek-cell on a

microscopy-slide, and Prof. Derek Chetwynd, who allowed me to use some of his

specialized micro-precision equipment.

I would also like to express my thanks to Dr. Daciana Udrea, Prof. Evor

Hines and Dr. Mike Jennings for proof-reading my thesis.

Last but not least I would like to thank Dr. Daniel Valdes Amaro, Mr. Jop

Vlaskamp, Dr. Hammad Qureshi, Mr. Vinncent O’Sullivan, Dr. Mike Jennings

and Mr. Michal Rutkowski and Mrs. Agnieszka Rutkowska for their friendship,

which gave me necessary energy and distance in order not to become totally

absorbed by the problems I faced during my PhD.

Abstract iii

List of Publications v

Declaration of Originality vii

Acknowledgements viii

List of Tables xv

List of Figures xxv

List of Symbols xxxii

1 Introduction 1

1.1 Historic Context . . . 1

1.2 Benefits of Digital Holography . . . 5

1.3 Outline of Thesis . . . 7

2 Fundamentals of Optics and Interferometry 10 2.1 Light Waves . . . 10

2.2 Intensity . . . 14

2.3 Polarization Optics . . . 15

2.4 Interference of Light . . . 22

2.5 Coherence . . . 25

2.6 Digital Recording Devices . . . 32

2.7 Approximation of Light Propagation . . . 34

2.7.1 Rayleigh-Sommerfeld Diffraction Formula . . . 34

2.7.2 Fresnel-Approximation or Near-Field Diffraction . . . 36

2.7.3 Fraunhofer Approximation or Far Field Approximation . . 39

2.7.4 Definition of Sign of Phase . . . 41

2.8 Conclusion . . . 41

3 Reconstruction Methods 43

3.1 Introduction . . . 43

3.2 Rayleigh-Sommerfeld Diffraction Integral . . . 45

3.2.1 Zero-padding . . . 47

3.2.2 Numerical Lens . . . 48

3.3 DC-term and Twin Image Suppression . . . 52

3.3.1 Averaged Intensity Subtraction . . . 55

3.3.2 Subtraction of Reference-wave and Object-wave Intensity . 58 3.3.3 Filtering in the Fourier Domain . . . 60

3.3.4 Phase stepping . . . 62

3.4 Fresnel-Method . . . 71

3.5 Fourier-Method . . . 73

3.6 Conclusion . . . 76

4 Optical Parameters and Properties of Digital Holography 78 4.1 Introduction . . . 78

4.2 Parameters for Optical Setup . . . 79

4.2.1 Minimum Distance Plane Reference-wave . . . 79

4.2.2 Minimum Distance Spherical Reference-wave . . . 81

4.2.3 Minimum Distance for Reconstruction Validity Region . . 82

4.3 Field of View (FOV) . . . 83

4.4 Resolution . . . 84

4.4.1 Lateral Resolution . . . 85

4.4.2 Axial Resolution or Depth of Field . . . 90

4.4.3 Phase Resolution . . . 93

4.5 Speckle . . . 96

4.6 Modulation Transfer Function . . . 98

4.6.1 MTF of Pixel-size . . . 99

4.6.2 MTF for Lens-less and Image-Holography . . . 101

4.7 Properties of Digital Holography . . . 104

4.7.1 Storage of 3-Dimensional Information . . . 104

4.7.2 Homogenous Information Distribution . . . 105

4.7.3 Direct Phase Accessibility . . . 107

4.7.4 Resolution Beyond Rayleigh-criterion . . . 114

4.8 Conclusion . . . 117

5 Space bandwidth product (SBP) 118 5.1 Introduction . . . 118

5.2 Required SBP0 of The Recording Sensor . . . 121

5.2.1 Fresnel-hologram . . . 122

5.2.2 Fourier-hologram . . . 128

5.2.3 Image-Hologram . . . 132

5.3 SBP00 or Performance Capacity . . . 140

5.3.1 Fresnel-hologram . . . 141

5.3.4 Conclusion . . . 143

6 Resolution Improvement in Digital Holography 147 6.1 Introduction . . . 147

6.2 Synthetic Aperture Method . . . 153

6.2.1 Setup and Methodology . . . 153

6.2.2 Deformation Measurement . . . 156

6.2.3 Resolution Improvement . . . 166

6.2.4 Extended Depth of Field (EDOF) Method . . . 170

6.2.5 Variance Approach . . . 173

6.2.6 Gaussian Fitting . . . 175

6.3 Sub-pixel Sampling Method . . . 179

6.3.1 Setup and Methodology . . . 179

6.3.2 USAF 1951 Test Target . . . 186

6.4 Combination of Synthetic Aperture and Sub-pixel Sampling Method189 6.4.1 Setup and Methodology . . . 189

6.4.2 Conclusion . . . 194

7 Digital Holographic Microscope 197 7.1 Introduction . . . 197

7.2 Setup . . . 200

7.3 Optical Parameters of DHM . . . 203

7.3.1 Thin Lens Model . . . 203

7.3.2 Reconstruction Distance (d0) . . . 206

7.3.3 Magnification, Image-Size, and Field of View (FOV) . . . . 207

7.3.4 Depth of Field (DOF) . . . 208

7.3.5 Experimental Validation . . . 209

7.4 Subtraction of Additional Spherical Phase Term . . . 210

7.5 Resolution Improvement . . . 213

7.6 Extended Depth of Field (EDOF) and 2D Refractive Index Dis-tribution . . . 214

7.7 Conclusion . . . 218

8 Conclusions and Future Work 220 8.1 Conclusion . . . 220

8.2 Future Work . . . 225

8.2.1 Software . . . 225

8.2.2 Hardware . . . 226

References 228

A Lens Equation 240

A.1 Derivation of Fibre Point Source Distance df band Projected Fibre

Point Source Distancedr . . . 240

A.2 Numerical Phase-Function of a Lens . . . 247

B Matlab Functions 249 C Solid-works Drawing 254 C.1 Solidworks assembly drawing for digital holographic microscope . 255 C.2 Solidworks assembly drawing for box covering the optical table . . 256

C.3 Solidworks assembly drawing holder for ‘Physik Instrumente’ x-y traverse . . . 257

C.4 Solidworks assembly drawing holder for fibre launcher . . . 258

C.5 Solidworks assembly drawing laser shutter . . . 259

D Data-Sheets of Optical Elements and Instruments used 260 D.1 Data-Sheet Pixelfly qe Camera . . . 261

D.2 Data-Sheet Piezo-Actuator . . . 262

D.3 Data-Sheet Wave-Plates . . . 263

D.4 Data-Sheet PI M-150.11 Stage . . . 264

D.5 Data-Sheet 6.6MP-CMOS sensor . . . 265

1.1 Measurement techniques for optical metrology (λ=632.8 nm), adapted

from Braunecker et al. (2008) . . . 6

2.1 Different types of polarization . . . 16

2.2 Jones-Matrix of polarization optical elements . . . 21

2.3 Different light sources used in interferometry and their coherence properties . . . 30

2.4 Comparison CCD vs. CMOS . . . 33

3.1 DC-term and twin-image suppression . . . 54

3.2 Comparison of termination speed . . . 57

3.3 Combination of λ/4- andλ/2-plate and the resulting phase step . 67 3.4 Evaluation of the phase step data . . . 68

3.5 Summary: Comparison reconstruction methods . . . 77

4.1 Source and corresponding parameters . . . 78

4.2 Speckle-size . . . 99

4.3 Theoretical and practically achieved resolution . . . 116

6.1 Consequences on lateral and axial alignment . . . 160

6.2 Trend-line standard deviation of phase . . . 164

6.3 SNR spatial averaging method for intensity and double exposure

phase map . . . 166

6.4 Theoretical and practically achieved resolution . . . 167

6.5 Gaussian coefficients and R-squared values . . . 176

6.6 Theoretical and practically achieved resolution . . . 188

6.7 Theoretical and practically achieved resolution . . . 192

6.8 SBP comparison . . . 193

7.1 Evaluation of the experimentally obtained data . . . 210

1.1 (a) Dennis Gabor taken from Nobel−prize. (2010), (b) Joseph W.

Goodman taken from Stanford University. (2010) . . . 1

1.2 (a) Recording and , (b) reconstruction of a hologram . . . 2

2.1 Propagation of electromagnetic light wave . . . 11

2.2 Transformation of cartesian coordinates in polar coordinates . . . 14

2.3 (a) Reflection and refraction between two media of refractive index

n1 and n2 , (b) Power reflectance R versus incident angle ϑ for

n1 = 1 and n2(BK7) = 1.517 . . . 17

2.4 Passage of linearly polarized light trough a half-wave retarder . . 19

2.5 Decomposition of wave vectors . . . 22

2.6 Creation of a photon by electron-jump from energy-level E1 toE2 25

2.7 (a) Interferometric setup for measuring the coherence length lc,

(b) reduced contrast for increasing differences between the path

lengths of the two interfering beams . . . 27

2.8 Young’s double slit experiment . . . 29

2.9 Intensity profiles for, (a) g=0.4 mm, (b) g=0.6 mm, and (c) g=1

mm . . . 29

2.10 Sketch of working principle of, (a) a CCD sensor, and (b) a CMOS

sensor . . . 32

2.11 Propagation of light from the object-plane to the hologram-plane 34

2.12 (a) Distances and corresponding wave-front for different

propaga-tion methods, (b) diffracpropaga-tion pattern of a rectangular aperture at

different distances . . . 40

2.13 Spherical-wave and plane-wave with (a) positive phase and (b)

negative phase . . . 41

3.1 Nomenclature of coordinates used for the holographic recording

and reconstruction process . . . 44

3.2 Optical model for Rayleigh-Sommerfeld Convolution integral . . . 46

3.3 (a) Recorded 2800x2800 zero-padded intensity hologram (b)

recon-struction with phase 2800x2800 pixels, (c) phase for a parabolic

lens, (d) phase for a spherical lens, (e) corresponding

reconstruc-tion of (c) by applying a shifted transfer-funcreconstruc-tion and Γ = 0.7

(1392x1392), (f) corresponding reconstruction of (d) with analogue

parameters . . . 47

3.4 Transfer-function: (a) modulus and (b) phase for at d’=120 mm;

(c) modulus and (d) phase for d’=67.3 mm . . . 50

3.5 Cascaded reconstruction in two steps . . . 51

3.6 (a) Amplitude transmission-function for sinusoidal amplitude

grat-ing, (b) Fraunhofer diffraction pattern . . . 53

3.7 (a) Amplitude transmission-function for averaged intensity

sub-tracted sinusoidal amplitude grating, (b) corresponding

Fraun-hofer diffraction pattern . . . 55

3.8 Influence of different ratios between reference- and object-wave

amplitude . . . 56

term suppression, (c) average intensity subtraction, (d) inverted

median filter, (e) sliding window operation, (f) subtraction of

ref-erence and object-beam . . . 59

3.10 (a) Numerical reconstruction after average-value subtraction, (b)

Numerical reconstruction with the inverted median filtered image,

(c) Numerical reconstruction with sliding window operation,(d)

Reconstruction by subtraction of reference and object-wave . . . . 60

3.11 (a) Amplitude transmission-function for saw tooth phase, (b)

cor-responding Fraunhofer diffraction pattern . . . 61

3.12 (a) Hologram in Fourier-domain, (b) blocked DC-term and

twin-image in Fourier-domain, (c) phase of resulting Fourier filtered

hologram, (d) numerical reconstruction . . . 62

3.13 (a) Setup with wave retarder plates, (b) setup with piezo-driven

mirror M1 . . . 63

3.14 Accuracy performance of Cai’s and Carr´e’s Methods . . . 67

3.15 Phase step accuracy obtained by (a) a wave-retarder plates and

(b) a piezo-driven mirror . . . 69

3.16 (a) Setup, (b) numerical reconstruction of the intensity hologram,

(c) numerical reconstruction of the phase hologram . . . 70

3.17 (a) Optical model for Fresnel method, (b) numerical reconstruction

utilizing Fresnel-method . . . 71

3.18 (a) Optical model for Fourier method, (b) numerically reconstructed

Fourier-hologram . . . 73

4.1 Geometry of interfering object and reference-wave when recording

a Fresnel-hologram . . . 80

4.2 Resulting intensity pattern due to interference of spherical

reference-and object-wave . . . 81

4.3 Lens field of view . . . 84

4.4 Geometry of plus first, minus first and zeroth diffraction order

caused by a binary grating . . . 86

4.5 Geometry circular aperture, (b) cross-section for resulting

diffrac-tion pattern . . . 87

4.6 (a) Geometry rectangular aperture, (b) cross-section in x”-direction

for resulting diffraction pattern . . . 88

4.7 (a) Normalized image intensity for two point sources of same

bright-ness and changing phase difference, which are separated by the

Rayleigh distance with respect to a rectangular aperture, (b)

nor-malized intensity in dependence of the phase difference . . . 89

4.8 (a) Geometric-optical DOF, (b) Geometric optical DOF and

wave-optical DOF . . . 90

4.9 (a) DOFλ and resolution for different wavelengths and NA, (b)

DOFg versus NA for ∆x0 = 6.54µm and λ= 632.8nm . . . 93

4.10 Effect of Quantization . . . 95

4.11 Axial and lateral speckle-size . . . 96

4.12 Imaging arrangement for (a) objective speckle and (b) subjective

speckle . . . 98

4.13 Sampling with different pixel-sizes . . . 100

4.14 Graph pixel MTF . . . 101

4.15 (a) Influence of νcutof f based on a rectangular aperture, (b)

Com-parison between lens-less holography with rectangular aperture

and image-plane holography with circular aperture . . . 103

reconstruction distance . . . 104

4.17 (a) Digital hologram (b) Numerical reconstruction suppressing DC-term (c) Masked hologram (d) Reconstruction (e) Negatively masked hologram (f) Reconstruction . . . 106

4.18 (a) Object under investigation, (b) cropped ambiguous exposure intensity reconstruction, (c) cropped unambiguous double-exposure phase reconstruction . . . 107

4.19 Setup geometry with illumination and observation point . . . 109

4.20 Setup geometry for measuring out-of plane displacement . . . 110

4.21 Setup geometry for measuring in-plane displacement . . . 111

4.22 (a) 5x5 Median filtered wrapped phase map, (b) Goldstein’s cut-line algorithm unwrapped phase map, (c) deformation map . . . . 112

4.23 Graph experimental data compared to model curve . . . 114

4.24 Recording of higher spatial frequencies . . . 114

4.25 (a) Hologram 3000x2208 pixels, (b) zero-padded hologram 6000x6000 pixels, (c) cropped reconstruction of (a), cropped reconstruction of (b) . . . 115

5.1 Adopted Figs. taken from Lohmann (1996) (a) SBP in the space-frequency domain, (b) SBP of (a) after Fresnel-transformation, (c) SBP of (a) after Fourier-transformation, (d) SBP of (a) after passage through lens . . . 120

5.2 Interference pattern caused by smallest resolvable object detail . . 122

5.3 Spatial frequency introduced by inclination of plane reference-wave 124

5.4 Diffracted cone of light from object coordinate xo to hologram-plane125

5.5 SBP of a Fresnel-hologram, (a) in-line arrangement, (b) off-line

arrangement . . . 128

5.6 Spatial frequency introduced by laterally offsetting the origin of

the spherical reference-wave . . . 129

5.7 SBP of a Frourier-hologram, (a) in-line arrangement, (b) off-line

arrangement . . . 131

5.8 Sketch of different planes involved in the image formation . . . 132

5.9 Convergent object illumination to suppress quadratic phase term . 137

5.10 SBP of an Image-hologram, (a) in-line arrangement, (b) off-line

arrangement . . . 139

6.1 (a) Sketch of setup for recording Fourier-holograms, (b) small

sec-tion of realized setup with camera and motorized x-y traverse . . 154

6.2 Segment of reconstructed hologram and double exposure phase

map both with profile line (a), (b) 3000x3000 pixels, (c), (d)

8800x8800 pixels and (e), (f) 3000x3000 pixels averaging approach 157

6.3 (a) Diffraction caused by collimated illumination of the sensor, (b)

sketch axial speckle de-correlation due to camera and/or curved

sensor, (c) phase offset caused by axial displacement between the

two double exposure camera positions . . . 158

6.4 Flow-chart of the spatial averaging approach applied to double

exposure holography . . . 161

6.5 (a) Phase error for vertical cutline of area under investigation, (b)

standard deviation of double exposure phase maps for adjacent

and furthest distant holograms including trend-line . . . 163

their position, (b) SNR for phase map versus number of images

and their position . . . 165

6.7 Hologram at recording distance of 295 mm for (a) 3000x3000 pixel,

(b) 8000x8000 pixel and their reconstructions (c) and (d),

respec-tively . . . 168

6.8 Region of interest including cross-section for (a) 3000x3000 pixel

hologram, (b) 8000x8000 pixel hologram . . . 169

6.9 (a) USAF 1951 test-target result with the spatial averaging

ap-proach, (b) displaying the same section for the twelve reconstructed

holograms . . . 170

6.10 Region of interest including cross-section for reconstruction

ob-tained from a 8000x8000 pixels zero-padded single hologram (3000x2208

pixels) . . . 171

6.11 (a) Intensity reconstruction for d0 = 728 mm, (b) intensity

recon-struction for d0 = 735 mm . . . 172

6.12 Topology map . . . 174

6.13 (a) Variance plot for both boxed areas shown in Fig. 6.11 including

Gaussian curve fitting, (b) Gaussian curve fitting for different WS

and comparison polynomial fitting for WS of 10x10 pixels . . . 176

6.14 Histograms for obtained topology map with, (a) traditional

vari-ance approach, (b) polynomial interpolation . . . 177

6.15 Cross-sections for indicated lines shown in Fig. 6.12, (a) A-A’, (b)

B-B’ . . . 178

6.16 Sketch of schematic setup . . . 179

6.17 Sketch of original camera pixel and schemata of four position

move-ment in order to obtain sub-pixel resolution . . . 180

6.18 Graph comparison of MTF sensor for the normal pixel-size and

half the pixel-size employing the sub-pixel sampling method . . . 182

6.19 Combination procedure to obtain a sub-pixel hologram . . . 182

6.20 Phase-hologram and reconstruction at 191 mm recording distance

for (a) normal hologram with 3.5µm pixel-size, (b) sub-pixel

holo-gram without phase correction and (c) sub-pixel holoholo-gram with

phase correction . . . 183

6.21 Sketch of phase-correction procedure . . . 184

6.22 Region of interest for intensity reconstruction and their profile lines

for (a) 300 mm recording distance with 3.5 µm pixel-size, (b)

sub-pixel hologram 191 mm recording distance . . . 185

6.23 Segment of double exposure phase maps for 191 mm recording

distance (a) normal hologram with 3.5µm pixel-size, (b) sub-pixel

hologram with 1.75 µm pixel-size . . . 185

6.24 Recording setup for determination of optical resolution . . . 186

6.25 (a) Intensity hologram 3000x3000 pixels, (b) modulus of calculated

complex object-wave 6000x6000 pixels, numerical reconstructions

(c) without sub-pixel sampling method, (d) with sub-pixel

sam-pling method, (e) and (f) corresponding areas of interest to

deter-mine smallest resolvable element . . . 187

6.26 Camera movement . . . 189

6.27 Flow chart combination procedure . . . 190

6.28 (a) Intensity hologram 3000x3000 pixels with 3.5 µm pixel-size at

295 mm, (b) and (c) phase and modulus of hologram 10040x10040

pixels with 1.75µm pixel-size at 185.25 mm . . . 191

6.28, (c), (d) region of interest to evaluate the resolution obtained

for both reconstructions, respectively . . . 192

7.1 Comparison of, (a) DHM results with 20xMO (NA=0.40) intensity

reconstruction and (b) phase reconstruction, (c) conventional

mi-croscope Polyvar 50xMO (NA=0.85) , (d) Zernike phase contrast

microscope Olympus IX51 40xMO (NA=0.60) . . . 199

7.2 (a) Schematic sketch of DHM setup, (b) practically realized setup 202

7.3 Specified and normalized dimensions for microscope objective . . 203

7.4 Imaging and reconstruction process . . . 205

7.5 Comparison of experimentally obtained data and calculated data

for, (a) reconstruction distance (d0), (b) magnification obtained for

the numerically reconstructed hologram (Γ00), (c) image-size in the

reconstruction-plane (y”), (d) object-size (y) . . . 209

7.6 (a) Hologram of human cheek-cell, (b) reconstructed phase

with-out suppression of spherical phase term , (c) numerically

calcu-lated spherical correcting wavefront, (d) reconstructed phase with

suppression of spherical phase term . . . 212

7.7 Intensity and phase reconstruction at d0 =91 mm for (a) and (b)

3000x3000 pixels with ∆x0 = 3.5µm , (c) and (d) 8805x8805 pixels

with ∆x0 = 1.75µm . . . 214

7.8 Reconstruction at (a) 91 mm and (b) 101 mm . . . 215

7.9 (a) Intensity reconstruction, (b) frequency filtered intensity

recon-struction, (c) EDOF map, (d) topology map unfiltered, (e)

topol-ogy map median filtered, (f) 2D refractive index distribution . . . 217

A.1 Passage of light through a plano-convex lens . . . 241

A.2 Parallel incident beams on a thin lens with ϑ the angle of incidence 246

A.3 Graphs dependence and accurracy of (a) df b and (b) dr on the

angle κT and the chosen calculation model . . . 247

B.1 Matlab function: Fresnel-method. . . 249

B.2 Matlab function: convolution. . . 250

B.3 Matlab function: fourierfocus. . . 251

B.4 Matlab function: phasemap cai. . . 252

B.5 Matlab function: gaussfit. . . 253

Symbol Name Definition

As orthogonal amplitude

Ap parallel amplitude

Ax vertical amplitude

Ay horizontal amplitude

α interference angle

α0 auxiliary variable α0 = πX_{λr}0x00

b distance between first and zeroth order

β0 auxiliary variable β0 = πY_{λr}0y00

β solid angle between z-y plane and~k;

solid angle between refracted ray of light and

normal vector

c speed of light in medium √1

µ =

c0 n

c0 speed of light in vacuum; √_{µ}1_{0}_{}_{0}

CCD charge coupled device

CMOS complementary metal-oxide semiconductor

d recording distance

d’ reconstruction distance

d1 object-lens distance _{d}f d_{2}_{−}2_{f}

d2 lens-image distance _{d}f d_{1}_{−}1_{f}

˜

d2 lens-hologram distance

df b fibre-lens distance

dimage distance between hologram and image d˜2−d2

do object distance

dr virtual reference source point distance to lens

dref reference source point distance to hologram

dR2 distance first lens surface to centre of

curva-ture

dsp lat lateral speckle size λ_{ζ}

dsp axial axial speckle size 4_{ζ}λ2

D lens-diameter

DHM digital holographic microscope

DOF depth of field seeldf

δ smallest resolvable object detail

δx0,δy0 size of intensity pattern in hologram-plane

∆νc spectral width _{τ}1

c

∆ϕ phase difference, phase delay ϕ2−ϕ1

∆ϕmax maximum phase resolution π_{2} −arccos_{l}_{grey}2

∆ϕmin minimum phase resolution arccos

1− 2

lgrey

∆x0 pixel-size in x’ direction

∆y0 pixel-size in y’ direction or lateral image

off-set for DHM

∆x00 pixel-size in x” direction _{N}λd_{∆}_{x}0 (for Fresnel)

∆y00 pixel-size in y” direction ∆y0 (for Rayleigh-Sommerfeld)

E energy

absolute permittivity or

0 electric constant; 8.85418·10−12As/V m

r relative permittivity _{}

0

ε diffraction angle

η diffraction efficiency

ηf ill fill-factor

ηSBP setup efficiency parameter _{SBP}SBP_{optimum}

f, f’ focal length in object and image plane h(n−1)_{R}1

1 + 1 R2 +

zLens(n−1) R1R2n

i

F focal point

F Fourier Transform

F−1 _{inverse Fourier Transform}

~

F force

FFT Fast Fourier Transform

ϕ,φ phase angle

g period of sinusoidal interference pattern

γ solid angle between z-x plane and~k

γ12 complex degree of coherence √ Γ12(r1,r2,τ)

Γ11(r1,0)Γ22(r2,τ)

Γ0 magnification y_{y}0

Γ00 magnification in reconstruction plane

Γ12 mutual coherence function

D p

I1(r1, t+τ)

p

I2(r2, t)

∗E

Γ11 intensity first pinhole I1(r, t)

Γ22 intensity second pinhole I2(r, t+τ)

h Planck’s constant; 6.626068·10−34_{m}2_{kg/s}

hd impulse-response of free space propagation

Hd transfer-function of free space propagation

~

H magnetic field strength

HH’ principle planes of a lens

I intensity

I imaginary part of complex number

IFTA fast fourier transform algorithm

Im imaginary part

~j Jones Vector

J Jones Matrix

k wave number

~

K sensitivity vector 4_{λ}π sinα

κ incident lens internal angle for second lens

surface

κT angle of refracted light after passage through

lens

l counter

lc coherence length

lν counter in Fourier domain

L cavity length

Lc cantilever length

laser light amplification by stimulated emission of

radiation

λ wavelength

m counter

mν counter in Fourier domain

M pixel-number in y’-direction

M’ pixel-number in y’-direction after

zero-padding for DHM

µ absolute permeability µ0µr

µ0 permeability constant 4π·10−7NA−2

µr relative permeability mu_{µ}

0

n refractive index √µrr

N pixel number in x’-direction

NA numerical aperture nsinσmax

NF Fresnel-number NF = X

02 λd

Nh number of holograms

Nzero pixel number required to display δ Nzero= 2N

ν frequency of light _{λ}c

νsa sampling frequency _{∆}1_{x}0; _{∆}1_{y}0

νx spatial frequency in x-direction

νx max maximum spatial frequency in x-direction

ν_{x}0 spatial frequency in x’-direction

νy spatial frequency in y-direction

ν_{y}0 spatial frequency in y’-direction

∇ Nabla operator; _{∂x}∂ i+ _{∂y}∂ j+_{∂z}∂ k

o21 auxiliary variable

o32 auxiliary variable

o31 auxiliary variable

PBC polarization beam splitter cube

PSF point spread function

PSI phase stepping interferometry

q electric charge

QWP quarter wave plate

r (curvature) radius

R power reflectance Rs+Rp 2

R real part of complex number

R2 squared correlation coefficient

Rs power reflectance of s-component sin

2_{(}_{ϑ}
T−ϑ)
sin2_{(}_{ϑ}

T+ϑ)

Rp power reflectance of p-component tan

2_{(}_{ϑ}
T−ϑ)
tan2_{(}_{ϑ}_{T+}_{ϑ}_{)}

RE real part

RMS root mean square value

s modulation depth

~

S poyntingvector E~ ×H~

SBP space bandwidth product

SLM spatial light modulator

SNR signal to noise ratio 20 logX¯

σ

σ standard deviation

t time

T period

TE transfers electric component

TM transfers magnetic component

TIFF tag image file format

τc coherence time l_{c}c

ϑ incident surface angle

ϑBr Brewster angle atann_{n}2_{1}

ϑR reflection angle

ϑT transmission or refraction angle

θ rotation angle of polarization optical element

u(x),

u(x,y)

object-plane

u’(x’,y’)

u”(x”),

u”(x”,y”)

reconstruction-plane

v auxiliary variable

V contrast Imax−Imin

Imax+Imin

ω angular frequency 2πν

x, y coordinates in the object-plane

y1 maximum height at which fibre emitted light

strikes the lens

x’, y’ coordinates in the hologram-plane or lens

image-plane

x”, y” coordinates in the reconstruction-plane

X object-size

X’ sensor-size N∆x0

X” image-size

¯

X mean value of population

zlens lens thickness

zmax displacement applied to cantilever at Lc

### CHAPTER 1

### Introduction

### 1.1

### Historic Context

[image:34.595.380.504.367.555.2](a) (b)

Figure 1.1: (a) Dennis Gabor taken from Nobel−prize. (2010), (b) Joseph W. Good-man taken from Stanford University. (2010)

In 1948 the Hungarian born British scientist Dennis Gabor shown in Fig. 1.1(a)

developed the theoretical concept of holography [Gabor (1948) and Gabor (1949)].

He coined the word holography from the Greek word ‘holos’- whole and ‘graphein’

to write. Holography is based on a light source of sufficient coherence. It consists

of two stages, the recording of the hologram and reconstruction of the image.

Laser

L2 L1 M1

M2 BC M3

hologram object

beam splitter cube lenses microscope objective mirrors

L4

L3

- BC - L1, L2, L3, L4

- MO

- M1, M2, M3

BC reference-wave

object-wave

(a)

Laser M1

M2 BC

hologram virtual image

L4 L3

reference-wave

observer

[image:35.595.118.514.95.586.2](b)

Figure 1.2: (a) Recording and , (b) reconstruction of a hologram

The recording process is schematically shown in Fig. 1.2(a). The light is

split into an object and a reference-wave. The object scattered light overlaps

with the reference-wave in the hologram plane, where it is recorded on a light

sensitive media such as a photographic plate. The object’s amplitude and phase

diffrac-1. Introduction 3

tion grating, which when illuminated with the reference-wave reproduces the

object-wave, see Fig. 1.2(b). The validity of Gabor’s idea could be confirmed

by a number of scientists Rogers (1952), El-Sum and Kirkpatrick (1952) and

Lohmann (1956). However, the interest in optical holography declined after a

few years due to poor image quality. The low image quality was caused by two

effects. Firstly, the lack of a sufficient coherent light source and secondly, the

overlap of the desired image with the twin-image and the undiffracted light

re-sulting in the DC-term in the reconstruction. The invention of the pulsed

Ruby-laser by Maiman (1960) and the separation of the reconstructed image-terms

by an off-line setup developed by Leith and Upatnieks (1962) gave holography

the necessary tools to emerge as one of the most promising optical techniques

of the 20th century. This can be confirmed by various publications made

there-after such as Thompson (1978), Hariharan (1984) and Ostrovsky et al. (1991).

Different holographic applications could be established such as holographic

par-ticle image velocimetry by Trolinger et al. (1969), holographic tomography by

Sweeney and Vest (1973) and its most important application in interferometry

by Powell and Stetson (1965). In holographic interferometry two or more object

states are compared interferometrically. Whereas at least one object state must

be holographically recorded and reconstructed according to Collier et al. (1965).

Various applications of holographic interferometry were developed such as

vi-bration analysis [Powell and Stetson (1965)], deformation measurement [Haines

and Hildebrand (1966)] or determination of refractive index changes caused by a

change of pressure or temperature [Horman (1965)]. The fringe counting, initially

performed manually, could soon be replaced by computer algorithms Osten et al.

(1987). This algorithm included digitizing and quantizing the photographed

op-tical reconstruction, calculating the phase distribution by utilizing the geometric

displaying the result. The introduction of phase shifting by Kreis et al. (1981)

was a significant step forward in the computer aided fringe analysis. It was

now possible to measure and not to estimate the interference phase. Moreover,

the phase sign ambiguity could be resolved utilizing phase shifting algorithms.

Although the fringe analysis was left to the computer, one still needed to wet

chemically process the holographic plate. The focus was now set on replacing the

holographic plate by digital means. Two solutions for the digital recording of

in-terferograms could be developed namely Electron-Speckle-Pattern-Interferometry

(ESPI) and digital holography. The first is based on in-focus recording of the

object under investigation by a digital camera. Diffusely scattering object were

investigated which resulted in the recording of the in-focus image covered with

speckle. The grainy speckle pattern could easily be recorded by existing analog

cameras. Adding the intensity of two recorded speckle pattern under different

ob-ject states results in the creation of correlation fringes similar to the one observed

with holographic interferometry. This technique became known as

Electronic-Speckle-Pattern-Interferometry (ESPI) and enabled the computerised recording

and processing of interferograms. ESPI today is a well established measurement

tool for metrology and used in many applications. The major drawback of ESPI

is the phase ambiguity which could just be resolved by the introduction of phase

stepping by Creath (1985) and Stetson and Brohinsky (1985). Another

disad-vantages of ESPI is the quality demand on the optics involved in order to avoid

the introduction of any kind of aberrations to the recorded speckle field, which

otherwise will decrease the accuracy of the interferometric measurement.

Con-trary to ESPI, digital holography enables lens-less recording, which reduces the

cost of the optical setup. Moreover, it permits phase determination by

record-ing a srecord-ingle off-line hologram. This results in less experimental effort and offers

1. Introduction 5

reconstructed utilizing computer aided methods by J. W. Goodman, shown in

Fig. 1.1(b), and R.W. Lawrence Goodman and Lawrence (1967). Hence

dig-ital holography is older than ESPI. A lens-less Fourier-hologram was recorded

on a vidicon-detector, consisting of a photoconductive surface scanned by an

electron-beam. The output of the vidicon was sampled in a 256x256 array with

a quantization of eight grey levels. This was the starting point of digital

holog-raphy, but it still took some decades until results of sufficient resolution became

available. Only in the last two decades digital holography has received more and

more importance which is strongly linked with the rapid development of digital

recording devices such as Charge-Coupled-Devices (CCD) and

Complementary-Metal−OxideSemiconductor (CMOS) cameras.

### 1.2

### Benefits of Digital Holography

Digital holography offers a higher degree of freedom for data acquisition and

processing than optical holography does. Moreover, it enables the direct

recon-struction of the phase without the need to apply phase stepping. This property

is beneficial for the investigation of dynamic events. It was first demonstrated

by Schnars (1994). Moreover, due to the numerical focusing a lens-less setup

is enabled, which reduces the cost and accuracy demand for an optical

sys-tem. On the contrary, in digital interferometry an optical system is needed,

which focuses on the object in order to record the phase correctly. A comparison

of digital holographic interferometry with various techniques applied in optical

metrology is shown in Table 1.1. Digital holographic interferometry can cover

a larger measurement range in the lateral dimension and likewise in the

lon-gitudinal dimension. If one wanted to cover the same range a combination of

Table 1.1: Measurement techniques for optical metrology (λ=632.8 nm), adapted from Braunecker et al. (2008)

Method Range lateral resolution RMS height resolution

Interferometry 1µm - 1000 mm 0.2 nm - 200 nm

Macroscopic fringe projection

10µm -2 m 10 µm - 1 mm

Microscopic fringe projection

1µm - 30 mm 0.1 µm - 10 µm

Confocal

micro-scope

0.5 µm - 30 mm 10 nm - 10 µm

White light inter-ferometry

0.7 µm - 5 mm 1 nm - 10 µm

Stylus instrument 100 nm - 100 mm 0.5 nm - 10 µm

Scattering 100 µm - 100 mm 0.5 nm - 10 nm

Digital holographic interferometry

500 nm - 1 m 1 nm - 50 mm

can be obtained by combing digital holography with microscopy demonstrated

in Osten (2006a). Kuehn et al. (2008) demonstrated a height resolution in the

sub-nanometer regime utilizing digital holography. A large range of measured

object height can be obtained applying holographic multi-wavelength contouring

demonstrated by Wagner et al. (2000). Moreover, the numerical reconstruction in

digital holography enables one to reduce wave-aberration effects as demonstrated

in Colomb et al. (2006). Additional benefits of digital holography are discussed

1. Introduction 7

Nevertheless, digital holography still suffers from low spatial resolution

typ-ically (100 lp/mm) in comparison to a photographic film (3000 lp/mm) used in

optical holography. This restricts the angle between object and reference-wave

and hence the object-size and resolution obtained.

### 1.3

### Outline of Thesis

This thesis focuses on the image quality and resolution improvement of

dig-ital holograms. The lateral resolution improvement will be demonstrated on

the USAF 1951 test-target, whereas the phase resolution improvement will be

demonstrated utilizing the standard deviation, which corresponds to the phase

measurement uncertainty. Double exposure phase maps will be used to prove the

phase improvement. The image quality improvement will be demonstrated by a

reduced Signal to Noise Ratio (SNR) applied to intensity reconstruction.

In Chapter 2the optical foundation in order understand the concept of dig-ital holography is represented. The relevant optical terminologies to describe the

recording and reconstruction process of digital holograms are explained. Terms

which refer to the recording process such as coherence, interference, intensity

and the working principle of digital recording devices are explained. The most

commonly applied diffraction models used for the numerical reconstruction of

digitally recorded holograms are discussed.

Chapter 3 represents the digitized numerical reconstruction methods. Ap-proaches to suppress the DC-term and the twin-image are introduced and

illus-tratively demonstrated by examples. Novel numerical methods developed by the

InChapter 4optical parameters of the reconstructed hologram are discussed. Rayleigh’s and Abbe’s resolution criteria including their valid application is

rep-resented. The derivation of optical parameters is presented, which is intended to

improve the readers understanding. Moreover, properties of digital holography

are represented, which in comparison to optical holography and other optical

techniques outline the benefits of digital holography.

Chapter 5 is devoted to the Space-bandwidth-product (SBP), which repre-sents an important parameter for the evaluation of optical systems. The required

SBP in the recording process and the SBP obtained in the reconstruction process

in respect to the hologram type is calculated. The work conducted by Lohmann

(1967) and Xu et al. (2005) was extended to cover in-line and off-line

config-urations of Fresnel-hologram, Fourier-hologram and Image-plane hologram. A

comparison of the performance of the three hologram types is conducted, which

reveals important information for the correct choice of hologram-type in respect

to the requirement.

Chapter 6represents the main work conducted during the PhD period. Res-olution improvement methods are represented and applied to digital holography.

The first resolution improvement approach is based on the synthetic aperture

method. Difficulties in conjunction with the recording of the object’s phase is

pointed out and possible solutions to overcome these difficulties are given.

More-over, the extended depth of field method is applied to the synthetic aperture

method, which to the author’s knowledge is novel. In addition to the synthetic

aperture approach, the novel sub-pixel sampling method in combination with

phase stepping is represented. The pixel sampling method is based on

1. Introduction 9

not least, both resolution improvement methods, synthetic-aperture and

sub-pixel sampling method, are combined.

In previous chapters all holograms described are recorded by a lens-less setup.

The holograms in Chapter 7 are recorded in combination with a

microscope-objective to improve the resolution in order to investigate object details in the

sub-micrometer region. Important steps for the recording process are highlighted

and the optical parameters obtained in the reconstruction are represented.

Fur-thermore, the resolution improvement methods discussed in Chapter 6 are applied

to prove their validity in conjunction with a lens system. Moreover, a proof of

principle to obtain the two dimensional refractive index contribution of the object

in conjunction with the extended depth of field method is shown.

Chapter 8 concludes this thesis and discusses the implication of this work for future research.

Each chapter consists of an introduction, main body and conclusion. This

### Fundamentals of Optics and

### Interferometry

Presented in this chapter are the physical principles to understand the recording

and reconstruction process in holography. The recording process, in which

co-herent light beams are superimposed, is presented. Requirements will be

demon-strated and terms like coherence and laser will be explained. The reconstruction

process based on different diffraction models and their validity will be discussed

in the second part of this chapter. The fundamental knowledge presented is in

close correspondence with Goodman (1996), Saleh and Teich (1991) and Kreis

(2005).

### 2.1

### Light Waves

Light is a transverse, electromagnetic wave characterized by a time-varying

elec-tric and magnetic field. The nature of light can be mathematically described

by Maxwell’s equations. Maxwell’s equation for a homogenous, isotropic, non

2. Fundamentals of Optics and Interferometry 11

T

t E

[image:44.595.179.462.106.207.2]B

Figure 2.1: Propagation of electromagnetic light wave

conducting medium is:

∇ ×E~ =−µ∂ ~H ∂t

∇ ×H~ =∂ ~E ∂t

∇ · ~E = 0

∇ ·µ ~H= 0

(2.1)

Where E~ is the electric field strength and H~ the magnetic field strength. t

indicates time and ∇ is the Nabla-operator which can be described according to

Goodman (1996) by:

∇= ∂

∂xˆi+ ∂ ∂yˆj+

∂ ∂z

ˆ

k (2.2)

and µ denote the permittivity and the permeability of the medium in which

the light wave propagates. The permittivity can be described by the product

of the relative permittivity and the permittivity in a vacuum ( = 0r). The

permeability can be calculated in an analogous manner (µ=µ0µr). Both, electric

and magnetic field, are perpendicularly orientated to each other, as shown in Fig.

2.1. A detailed derivation from Maxwell’s equation to the wave equation can be

found in Goodman (1996). The wave equation obtained is:

∇2_{~}_{u}_{−} 1

c2

∂2_{~}_{u}

The wave-equation for the electric field strengthE~ and the magnetic field strength

~

H are identical. Both, E~ and H~, have therefore been replaced by the symbolic

vector ~u, as shown in Eq. 2.3. c is the speed of light in the medium. c can be

calculated using:

c= c0

n (2.4)

Where the refractive index n can be expressed by a combination of relative

per-mittivity r and relative permeability µr. For most materials µr at optical

fre-quencies is close to one.

n=√µrr ≈

√

r (2.5)

Therefore, the electrical field can be considered to be the predominant interaction

component of the electro-magnetic wave with the material. Thus~ucan be treated

as a replacement for the electric field strength E~. Light waves are transverse

waves oscillating perpendicular to the direction of propagation and are therefore described in vector notation. For most applications it is not necessary to use

the full vector description of the field. A light wave oscillating in a single plane,

namely a linearly polarized light wave, which propagates in the z-direction can

be described in scalar notation as:

u(z, t) =A0cos (kz−ωt+ϕ0) (2.6)

Eq. 2.6 is also known as the harmonic wave equation. k is the wave number,

which depends on the wavelength of light λ.

k = 2π

2. Fundamentals of Optics and Interferometry 13

ω is theangular frequency, which is related to the frequency of light ν by

ω = 2πν = 2π

T (2.8)

with

ν = c

λ (2.9)

The time for a whole 2π cycle is called the period T. Taking into account the

substitutions made in Eq. 2.7 and Eq. 2.8 results in a harmonic wave-equation:

u(z, t) =u0cos

2π

λ z−

2π T t+ϕ0

(2.10)

The harmonic wave equation can be expressed in complex notation by applying

Euler’s formula.

exp (iα) = cosα+isinα (2.11)

It is important to consider that only the real or imaginary part matches with Eq.

2.6 and hence makes physically sense.

u(z, t) = Re{A0exp [i(kz−ωt+ϕ0)]} (2.12)

This complex notation offers some advantages in terms of expressing the phase or

to modulate the phase as we will see in Chapter 3. An alternative waveform often

used to describe the nature of light is thespherical wave. In mathematical terms

the coordinates are changed to polar coordinates x = rsinβ, y = rcosβsinγ

### r

### b

### g

### y

### x

[image:47.595.218.420.103.294.2]### z

Figure 2.2: Transformation of cartesian coordinates in polar coordinates

Due to its geometry a spherical wave does not dependent on β and γ. Thus

the scalar wave equation becomes

1

r ∂2

∂r2ru− 1

c2

∂2_{u}

∂t2 = 0 (2.13)

### 2.2

### Intensity

The phase ϕof the wave described in Eq. 2.12:

ϕ= 2π

λ z−ωt+ϕ0 (2.14)

is proportional to the wavelength of light λ = _{ν}c. The wavelength of visible

light ranges from 400 nm to 800 nm. The corresponding frequency ranges from

7.5·1014 _{Hz to 3}_{.}_{7}_{·}_{10}14 _{Hz. Light sensor such as the eye, photodiode, CCD or}

CMOS are not able to detect such high frequencies. The only quantity which can

be measured is the intensity. The intensity observed correlates to the sum of the

2. Fundamentals of Optics and Interferometry 15

field (ρmag) ∗:

ρE =ρel+ρmag = 1

2E

2

+ 1

2µH

2

(2.15)

WithH = _{cµ}E and c2 = _{µ}1 follows:

ρE = 1

2E

2_{+} 1

2E

2 _{=}_{E}2 _{(2.16)}

The electromagnetic energy is recorded in energy packets called photons. The

photon energy Epho is converted by the photoelectric effect into free electrons

with a certain kinetic energy Ekin.

Epho =WA+Ekin (2.17)

Where WA is the material dependent energy gap which needs to be exceeded

in order to generate free electrons. The intensity recorded is hence proportional

to E2_{. The frequency of visible light is of such large magnitude (10}14_{) that even}

the fastest high speed cameras (≈ 105 _{Hz) are too slow to resolve a time period}

of light. Hence the recorded intensity represents the integration of E2 _{quantities.}

### 2.3

### Polarization Optics

The light wave is a transversal wave, which can have different planes of oscillation.

In each plane perpendicular to the direction of propagation the vector of the

electric field strength E~ and in analogue manner the magnetic field strength H~

follows a specific path Haferkorn (2003). The path may be describe a straight line,

a circle or an ellipse. These different states of wave propagation can be described

as linearly, circularly or elliptically polarized light. They can be produced by two

Table 2.1: Different types of polarization

phase difference

∆ϕ Ax 6= Ay Ax =Ay

0 ◦ linearly polarized light diagonal linearly polarized

0 ◦ < ϕ <90◦ left elliptically polarized left elliptically polarized

ϕ= 90 ◦ left elliptically polarized left circularly polarized

90◦ < ϕ <180 ◦ left elliptically polarized left elliptically polarized

ϕ= 180◦ linearly polarized diagonal linearly polarized

180 ◦ < ϕ <270 ◦ right elliptically polarized right elliptically polarized

270◦ right elliptically polarized right circularly polarized

270 ◦ < ϕ <360 ◦ right elliptically polarized right elliptically polarized

linearly polarized wave-trains, which are orientated perpendicular to each other.

The resulting states of polarization for various combinations of two perpendicular

wave-trains with amplitude Ax and Ay and phase difference ∆ϕ between both

wave-trains are shown Table 2.1.

There are different possibilities to obtain linearly polarized light. One way is

to generate linearly polarized light within a laser cavity utilizing the Brewster

window. Fig. 2.3(a) shows the light path starting from the incident wave on a

dielectric medium, which is then split into a reflected wave, indicated by subscript

R, and a refracted wave, indicated by subscript T.

If the sum of the reflection angle ϑR and the refraction angle ϑT is 90 ◦

only the s-component is reflected. The abbreviation ‘s’ arises from the German

word ‘senkrecht’, which means orthogonal. It is also referred to as the transverse

electric (TE) or orthogonal component. The p-component, parallel component,

also known as the transverse magnetic (TM) passes through the glass without

experiencing any reflection loss. The power reflectance for the s-component Rs,

2. Fundamentals of Optics and Interferometry 17 As Ap q qR qT n incide t w

ave

fl cte_{d wav}
e
re e

n i t trasm t

ed

d e

refracte wav

n1 n2
AsR
AsT
ApT
incide
nt
wave
e
refl
cte_{d }
wa_{ve}

tr m t

d

ans it e

efr ted w

ve r ac a As Ap q qR qT n1 n2 AsR ApT AsT Ax Ay optically slow axis optically fast axis z Dj=p q1 q1 q2 (a)

0 pi/6 pi/3 pi/2

0 0.2 0.4 0.6 0.8 1

ϑ in rad

Power reflectance R

[image:50.595.208.433.102.482.2]R R R S P (b)

Figure 2.3: (a) Reflection and refraction between two media of refractive index n1 and

n2 , (b) Power reflectance R versus incident angle ϑ forn1 = 1 and n2(BK7) = 1.517

as:

Rs =

sin2(ϑT −ϑ) sin2(ϑT +ϑ)

Rp =

tan2_{(}_{ϑ}
T −ϑ)
tan2(ϑT +ϑ)

R= Rs+Rp 2

(2.18)

The power reflectance as a function of the incident angleϑfor BK7-glass is shown

in Fig. 2.3(b). The incident angle at which only the s-component is reflected is

calculated as follows:

n1sinϑBrn1 =n2sin (90◦−ϑBr) =⇒ ϑBr = atan

n2

n1

(2.19)

This relationship can be used to obtain polarized laser light. The vector normal

to the surface of the polarising glass (also known as the Brewster window) is

orientated at the Brewster-angle to the optical axis of the resonator. The

s-component is thus reflected orthogonally outside the laser cavity. Light within

the laser cavity is reflected back and forth. Therefore it passes through the

Brewster window several times, which results in laser light containing only the

p-component. Working with linearly-polarized light in interferometry offers a high

degree of freedom by adapting either intensity and/or polarization to different

situations. Moreover, the contrast of the interference pattern can be increased,

which results in an improved resolution for the reconstructed hologram, which

will be discussed in Section 4.4.3.

Sometimes the polarization state of one of the two interfering beams needs

to be changed in order to obtain matched polarization state of both beams.

Moreover, changing the polarization state enables the acquisition of more object

information, as shown in Whittaker et al. (1994). A change of polarization state

can be accomplished by the introduction of wave-retarder plates, which possess

a certain phase delay ∆ϕ between optically slow and optically fast axis. The

incident wave, shown in red in Fig. 2.4, oscillates on a plane at an angle θ1 to

the x-z plane. The amplitude of the incident wave can be split into its vertical

Ax (green) and horizontalAy (yellow) component. We assume that the optically

fast axis is vertically aligned as shown in Fig. 2.4. The horizontal component in

Fig. 2.4 is phase delayed by π (half-wave retarder) to the vertical component.

2. Fundamentals of Optics and Interferometry 19 As Ap q qR qT

i cid nt ve n e wa

fl cte
d wav_{e}
re e

tr na smi tedt r f c d

wa e e ra te

v

### n

1### n

_{2}

AsR

AsT

ApT

### n

### i e

### i c

### d

### nt

### a

### e

### w

### v

### ec

### refl

### ted

### wa

_{ve}

### tran

### smit

### ted

### refra

### cted

### wav

### e

### A

_{s}

### A

p### q

### q

R### q

T### n

_{1}

### n

_{2}

### A

_{sR}

### A

_{pT}

### A

_{sT}

### A

_{x}

### A

y### optically slow

### axis

### optically fast

### axis

### z

q1 q1 q2### optically slow

### axis

### optically fast

### axis

### Dj=p

### A

x### A

y q1### z

q2Figure 2.4: Passage of linearly polarized light trough a half-wave retarder

The influence of wave-retarders on the incident beam can be predicted

utiliz-ingJones-Calculus. A wave-retarder with the phase delay ∆ϕcan be described by the following Jones-Matrix.

J=

1 0

0 exp (−i∆ϕ)

(2.20)

The incident light maintains a certain angle θ1 to the x-z plane. Its Jones-vector

is therefore obtained by its vertical and horizontal projection.

cosθ1 sinθ1

(2.21)

A possible rotation angle θ between wave-retarder and the x-z-plane is

ac-counted for by applying the rotation-matrixR(θ) to J.

R(θ) =

cosθ sinθ

−sinθ cosθ

The rotated wave-retarder becomes:

JR(θ) = R (θ)

−1

JR (θ) =

cosθ −sinθ

sinθ cosθ

1 0

0 exp (−i∆ϕ)

cosθ sinθ

−sinθ cosθ

=

cos2_{θ}_{+ sin}2_{θ}_{exp (}_{−}_{i}_{∆}_{ϕ}_{)} 1

2sin 2θ[1−exp (−i∆ϕ)] 1

2sin 2θ[1−exp (−i∆ϕ)] sin

2_{θ}_{+ cos}2_{θ}_{exp (}_{−}_{i}_{∆}_{ϕ}_{)}

(2.23)

The rotated Jones-matrices of the most commonly applied polarization optical

elements are shown in Table 2.2. It needs to be emphasized that θ of each

individual element does not necessarily need to be the same. The influence of

the polarization-optical element on the incident wave can be calculated using:

~jout = JR~jin (2.24)

Polarization optical elements in conjunction with digital holography will be used

2. Fundamentals of Optics and Interferometry 21

Table 2.2: Jones-Matrix of polarization optical elements

Polar-ization ele-ment Jones-Matrix

Rotated Jones Matrix Application

Rotated polarizer 0 0 0 1

cos2_{θ} 1
2sin 2θ
1

2sin 2θ sin 2θ Generate lin-early polarized light, polariscope, photography Rotated half-wave retarder 1 0

0 −1

cos 2θ1 sin 2θ1

sin 2θ1 −cos2θ1 Phase stepping, polarization alignment in interferometry Rotated quarter-wave retarder 1 0

0 −i

cos2_{θ}

1−isin2θ1 1_{2}(1 +i) sin 2θ1
1

2(1 +i) sin 2θ1 sin
2_{θ}

1−icos2θ1

Phase stepping,

polarization

align-ment in

Michelson-interferometer,

polariscope,

op-tical isolator in

conjunction with a

polarizer Dielectric mirror 1 0

0 −1

cos 2θ1 sin 2θ1

sin 2θ1 −cos 2θ1

Polarization

### 2.4

### Interference of Light

This section is focused on the explanation of the principles of interferometry.

The effect of interference occurs when two or more coherent light waves are

superimposed. Let us consider two waves with amplitudes A01 and A02. The

angular frequency ω and the polarization plane is assumed to be the same for

both waves. The different directions of propagation of both waves is accounted

for by the corresponding k-vector.

u1(r, t) = A01exp

h

i

~

k1r−ωt+ϕ1

i

u2(r, t) = A02exp

h

ik~2r−ωt+ϕ2

i (2.25)

### k

1### b

1### g

1### y

### x

### z

### k

2### b

2 [image:55.595.222.419.297.581.2]### g

22. Fundamentals of Optics and Interferometry 23

The interference can be described as:

(u1(r, t) +u2(r, t)) =A01exp

h

i~k1r−ωt+ϕ1

i

+A02exp

h

i~k2r−ωt+ϕ2

i

= exp (−iωt)

A01exp

n

i~k1r+ϕ1

o

+A02exp

n

i~k2r+ϕ2

o

= exp (−iωt) [A01exp (iφ1) +A02exp (iφ2)]

(2.26)

A possible combination of two different k-vectors is shown in Fig. 2.5. ~k1 and~k2

can be described as:

~k1 =

~kz cosβ1cosγ1

~k2 =

~kz cosβ2cosγ2

(2.27)

The recording media detects the intensity of both interfering waves, which is

defined as:

I(r) = |u1(r, t) +u2(r, t)|2

= (u1(r, t) +u2(r, t)) (u1(r, t) +u2(r, t))

∗

=A2_{01}+A2_{02}+A01A02exp (i(φ1−φ2))
+A01A02exp (−i(φ1−φ2))

=A2_{01}+A2_{02}+ 2A01A02cos (φ1−φ2)
=A2_{01}+A2_{02}+ 2A01A02cos (∆φ)

(2.28)

I(r) =A2_{01}+A2_{02}+ 2A01A02cos

kzr

1 cosβ1cosγ1

− 1

cosβ2cosγ2

+ ∆ϕ

(2.29)

One can distinguish between constructive and destructive interference.